The process is a bit complicated. An SD card (with at least 512MB of free space) must first be inserted into the Wii U, after which the system transfer option can be selected from the Wii U menu. The SD card is then moved to the Wii, where a "transfer application" is downloaded from the Wii Shop. Said app moves all of the Wii's data (save for pre-installed software) to the SD card. Once that's done, the card is popped back into the Wii U and, finally, the data is transferred to the new console.
Both the Wii and Wii U must be connected to the internet – presumably to verify the associated Wii user – and connected to a TV, meaning you'll either have to swap TV inputs or swap cables during the process. Data transferred includes WiiWare and Virtual Console games, save data for all games, DLC, Wii Points and Miis.
Once the transfer is complete, that's it. All data will be deleted from the Wii and moved to the Wii U, after which the data on the SD card is also erased. Data cannot be transferred from a Wii U to a Wii. We'll have a detailed video of the process for you when the Wii U officially launches later this month.